These results are obtained via a real root (with an appropriate property) of the characteristic equation

Electronic Journal of Differential Equations, Vol. 2004(2004), No. 03, pp. 1–17.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

ASYMPTOTIC PROPERTIES, NONOSCILLATION, AND STABILITY FOR SCALAR FIRST ORDER LINEAR AUTONOMOUS NEUTRAL DELAY DIFFERENTIAL

EQUATIONS

CHRISTOS G. PHILOS & IOANNIS K. PURNARAS

Abstract. We study scalar first order linear autonomous neutral delay dif- ferential equations with distributed type delays. This article presents some new results on the asymptotic behavior, the nonoscillation and the stability.

These results are obtained via a real root (with an appropriate property) of the characteristic equation. Applications to the special cases such as (non-neutral) delay differential equations are also presented.

1. Introduction

Neutral delay differential equations are differential equations depending on past and present values, which involve derivatives with delays as well as the unknown function itself. Besides its theoretical interest, the study of such equations has great importance in various applications in natural sciences and technology. For the basic theory of neutral delay differential equations, the reader is referred to the books by Diekmannet al. [2], Hale [10], and Hale and Verduyn Lunel [11].

Driver, Sasser and Slater [6] have obtained some significant results on the as- ymptotic behavior, the nonoscillation and the stability for a first order linear delay differential equation with constant coefficients and one constant delay. These re- sults have been improved and extended by Philos [13] for first order linear delay differential equations in which the coefficients are periodic functions with a common period and the delays are constants and multiples of this period. The results in [6]

have also been improved and extended by Kordonis, Niyianni and Philos [12] for first order linear neutral delay differential equations with constant coefficients and constant delays. Philos and Purnaras [14] have studied the more general case of first order linear neutral delay differential equations with periodic coefficients and constant delays, where the coefficients have a common period and the delays are multiples of this period. The results in [14] contain especially those in [13] (in an improved version) as well as the ones given in [12]. Moreover, the results obtained by Graef and Qian [8] are also motivated by those in [6] and are closely related.

2000Mathematics Subject Classification. 34K11, 34K20, 34K25, 34K40.

Key words and phrases. Neutral differential equation, asymptotic behavior, nonoscillation, stability.

c

2004 Texas State University-San Marcos.

Submitted October 6, 2003. Published January 2, 2004.

1

For some related results we refer to the papers by Arino and Pituk [1], Driver [4], and Gy¨ori [9].

In [3], Driver studied first order linear autonomous delay differential equations with infinitely many distributed delays and obtained some important results on the asymptotic behavior, the nonoscillation and the stability. For previous related results we refer to the references cited in [3]. The results given in this paper are essentially motivated by the corresponding ones in [3] and the techniques applied in the present paper are originated in some of the methods used in [3].

This paper deals with the asymptotic behavior, the nonoscillation and the sta- bility for scalar first order linear autonomous neutral delay differential equations with distributed type delays. A basic asymptotic criterion is established. Also, a nonoscillation result is given. Moreover, a useful estimate of the solutions is ob- tained and a stability criterion is derived. Our results are obtained by the use of a real root (with an appropriate property) of the corresponding characteristic equation. The results given here can be applied to the corresponding non-neutral equations. An application of our results to the special case of (non-neutral) delay differential equations leads to an improved version of some of the results given by Driver in [3].

Recently, a very interesting article has been published by Frasson and Verduyn Lunel [7] concerning the large time behaviour of linear functional differential equa- tions. It is shown there that the spectral theory for linear autonomous as well as periodic functional differential equations yields explicit formulas for the large time behaviour of solutions. The results in [7] are based on resolvent computations and Dunford calculus. Some known results (see [6, 12]) can be obtained as applications of the general results given in [7]. The work in [7] may be viewed as a gener- alization of previous works for first order scalar linear autonomous and periodic functional differential equations (see [3, 6, 12, 13, 14]). It must be noted that, in [3, 6, 12, 13, 14] as well as in the present paper, the method used in obtaining the results is very simple and is essentially based on elementary calculus.

Consider the neutral delay differential equation h

x(t) + Z 0

−σ

x(t+s)dζ(s)i0

= Z 0

−τ

x(t+s)dη(s), (1.1) whereσ andτ are positive constants, ζ andη are real-valued functions of bounded variation on the intervals [−σ,0] and [−τ,0] respectively, and the integrals are Riemann-Stieltjes integrals. It will be supposed that η is not constant on[−τ,0].

Set

r= max{σ, τ}.

Clearly,ris a positive constant.

As usual, a continuous real-valued function xdefined on the interval [−r,∞) is said to be asolution of the neutral delay differential equation (1.1) if the function x(t) +R0

−σx(t+s)dζ(s) is continuously differentiable fort≥0 andxsatisfies (1.1) for allt≥0.

In the sequel, byC([−r,0],R) we will denote the set of all continuous real-valued functions on the interval [−r,0]. This set is a Banach space endowed with the sup- normkφk= sup_t∈[−r,0]|φ(t)|.

It is well-known (see, for example, Diekmannet al. [2], Hale [10], or Hale and Verduyn Lunel [11]) that, for any given initial function φ in C([−r,0],R), there

exists a unique solutionxof the differential equation (1.1) which satisfies theinitial condition

x(t) =φ(t) fort∈[−r,0]; (1.2) this functionxwill be called the solution of theinitial problem (1.1)-(1.2) or, more briefly, the solution of (1.1)-(1.2).

Thecharacteristic equation of (1.1) is λh

1 + Z 0

−σ

e^λsdζ(s)i

= Z 0

−τ

e^λsdη(s). (1.3)

Throughout this paper, by V(ζ) we will denote the total variation function of ζ, which is defined on the interval [−σ,0] as follows: V(ζ)(−σ) = 0, andV(ζ)(s) is the total variation ofζon [−σ, s] for eachsin (−σ,0]. Also,V(η) will stand for the total variation function of η defined on the interval [−τ,0] by an analogous way:

V(η)(−τ) = 0, andV(η)(s) is equal to the total variation of η on [−τ, s] for each s∈(−τ,0]. Note that the functionsV(ζ) andV(η) are nonnegative and increasing on the intervals [−σ,0] and [−τ,0] respectively. Moreover, it must be noted that V(ζ) is identically zero on [−σ,0] ifζ is constant on this interval, and thatV(η) is not identically zero on the interval [−τ,0] (and so it is always not constant on [−τ,0]). It will be considered that the reader is familiar with the theory of functions of bounded variation and the theory of Riemann-Stieltjes integration.

To obtain the main results of this paper, we will make use of a real root λ0 of the characteristic equation (1.3) with the property

Z 0

−σ

[1 +|λ0|(−s)]e^λ0sdV(ζ)(s) + Z 0

−τ

(−s)e^λ0sdV(η)(s)<1. (1.4) Let us consider the special case of the (non-neutral) delay differential equation

x⁰(t) = Z 0

−τ

x(t+s)dη(s). (1.5)

This equation can be obtained (as a special case) from the differential equation (1.1), by choosingσto be an arbitrary positive constant withσ≤τand considering ζ to be any constant real-valued function on [−σ,0].

As it concerns the (non-neutral) delay differential equation (1.5), we have the constantτ in place ofr.

By a solution of (1.5), we mean a continuous real-valued function xdefined on the interval [−τ,∞), which is continuously differentiable on [0,∞) and satisfies (1.5) fort≥0. In the special case of (1.5), the Initial Condition (1.2) becomes

x(t) =φ(t) fort∈[−τ,0]. (1.6) Thecharacteristic equation of (1.5) is

λ= Z 0

−τ

e^λsdη(s). (1.7)

With respect to the (non-neutral) delay differential equation (1.5), we need a real rootλ0 of the characteristic equation (1.7) with the property

Z 0

−τ

(−s)e^λ0sdV(η)(s)<1. (1.8)

The notions of the stability, instability, uniform stability, asymptotic stability anduniform asymptotic stability of thetrivial solutionof a neutral (or non-neutral) delay differential equation will be considered in the usual sense (see, for example, Diekmannet al. [2], Hale [10], or Hale and Verduyn Lunel [11]; for the non-neutral case, see also Driver [5]). Note that, since the differential equation (1.1) (and, in particular, the differential equation (1.5)) is autonomous, the trivial solution of (1.1) (and, in particular, of (1.5)) is uniformly stable or uniformly asymptotically stable if and only if it is stable (at 0) or asymptotically stable (at 0) respectively.

Our main results are two theorems and two corollaries of the first of these theo- rems. The main results of the paper are stated in Section 2. The proof of the first theorem is given in Section 3, while the proof of the second theorem is presented in Section 4. Section 5 is devoted to the application of the main results to the special case of the (non-neutral) delay differential equation (1.5). Sufficient conditions for the characteristic equation (1.3) (and, in particular, for (1.7)) to have a real rootλ0

with the property (1.4) (and, in particular, with the property (1.8)) are obtained in Section 6.

2. Statement of the main results

Theorem 2.1 below is a basic asymptotic criterion for the solutions of the neutral delay differential equation (1.1).

Theorem 2.1. Let λ0 be a real root of the characteristic equation (1.3)with the property (1.4)and set

γ(λ0) = Z 0

−σ

[1−λ0(−s)]e^λ0sdζ(s) + Z 0

−τ

(−s)e^λ0sdη(s).

Then, for everyφ∈C([−r,0],R), the solution xof (1.1)-(1.2)satisfies

t→∞lim

e^−λ0tx(t)

= L(λ0;φ) 1 +γ(λ0), where

L(λ0;φ) =φ(0) + Z 0

−σ

h

φ(s)−λ0e^λ0s Z 0

s

e^−λ0uφ(u)dui dζ(s)

+ Z 0

−τ

e^λ0shZ 0 s

e^−λ0uφ(u)dui dη(s).

Note: Property (1.4) guarantees that 1 +γ(λ0)>0.

We immediately see that λ0 = 0 is a root of the characteristic equation (1.3) with the property (1.4) if and only if

Z 0

−τ

dη(s) = 0 and Z 0

−σ

dV(ζ)(s) + Z 0

−τ

(−s)dV(η)(s)<1, i.e. if and only if the following condition holds:

η(−τ) =η(0) and V(ζ)(0) + Z 0

−τ

(−s)dV(η)(s)<1. (2.1) Note that V(ζ)(0) is the total variation of ζ on the interval [−σ,0]. Thus, an application of Theorem 2.1 withλ₀= 0 leads to the following corollary.

Corollary 2.2. Let Condition (2.1)be satisfied. Then, for φ∈C([−r,0],R), the solution xof (1.1)-(1.2)satisfies

t→∞lim x(t) =φ(0) +R0

−σφ(s)dζ(s) +R0

−τ

R0

s φ(u)du dη(s) 1 + [ζ(0)−ζ(−σ)] +R0

−τ(−s)dη(s) . Note: The second assumption of (2.1) ensures that

1 + [ζ(0)−ζ(−σ)] + Z 0

−τ

(−s)dη(s)>0.

Another immediate consequence of Theorem 2.1 is the following result. As cus- tomary, a solution of (1.1) is said to be nonoscillatory if it is either eventually positive or eventually negative.

Corollary 2.3. Let λ₀ be a real root of the characteristic equation (1.3) with the property (1.4). Then, for any φ∈C([−r,0],R), the solution x of (1.1)-(1.2) will be nonoscillatory, except possibly if φis such that L(λ₀;φ) = 0, where L(λ₀;φ)is defined as in Theorem 2.1.

Consider a real root λ0 of (1.3) with the property (1.4) and, for any φ ∈ C([−r,0],R), let L(λ0;φ) be defined as in Theorem 2.1. Clearly, the operator L(λ0;·) is linear. Moreover, there exists a function φ0 ∈ C([−r,0],R) such that L(λ0;φ0)6= 0. Indeed, if we set

φ₀(t) =e^λ0t fort∈[−r,0], thenφ0∈C([−r,0],R) and we have

L(λ0;φ0)≡φ0(0) + Z 0

−σ

h

φ0(s)−λ0e^λ0s Z 0

s

e^−λ0uφ0(u)dui dζ(s)

+ Z 0

−τ

e^λ0shZ 0 s

e^−λ0uφ₀(u)dui dη(s)

= 1 + Z 0

−σ

e^λ0s−λ0e^λ0s(−s)

dζ(s) + Z 0

−τ

e^λ0s(−s)dη(s)

= 1 + Z 0

−σ

[1−λ0(−s)]e^λ0sdζ(s) + Z 0

−τ

(−s)e^λ0sdη(s)

= 1 +γ(λ0)>0,

where γ(λ₀) is defined as in Theorem 2.1. So, by the same method with the one that was used by Driver in [3] (see, also, Philos [13]), one can prove the following result, which can be considered as a complement of Corollary 2.3.

Let λ0 be a real root of the characteristic equation (1.3)with the property (1.4).

Moreover, for any φ ∈ C([−r,0],R), let L(λ0;φ) be defined as in Theorem 2.1.

Then the set of all functions φ ∈ C([−r,0],R) which satisfy L(λ0;φ) = 0 is a nowhere dense subset of the Banach spaceC([−r,0],R)(with the sup-norm).

The following theorem establishes an estimate for the solutions of the neutral delay differential equation (1.1) and, also, a stability criterion for the trivial solution of (1.1).

Theorem 2.4. Let λ0 be a real root of the characteristic equation (1.3)with the property (1.4). Consider γ(λ0)as in Theorem 2.1 and set

µ(λ₀) = Z 0

−σ

[1 +|λ0|(−s)]e^λ0sdV(ζ)(s) + Z 0

−τ

(−s)e^λ0sdV(η)(s).

Then, for anyφ∈C([−r,0],R), the solution xof (1.1)-(1.2)satisfies

|x(t)| ≤N(λ₀)kφke^λ0t for allt≥0, where

N(λ₀) = 1 +µ(λ₀) 1 +γ(λ0) +h

1 +1 +µ(λ₀) 1 +γ(λ0) i

µ(λ₀) max{1, e^λ0r}.

Here the constantN(λ0)is greater than 1. Moreover, the trivial solution of (1.1)is uniformly stable if λ0= 0, uniformly asymptotically stable ifλ0<0, and unstable if λ0>0.

Note that the criterion for the uniform stability stated in Theorem 2.4 can equiv- alently be formulated as follows:

The trivial solution of (1.1)is uniformly stable if Condition (2.1)holds.

3. Proof of Theorem 2.1 First of all, let us defineµ(λ0) as in Theorem 2.4, i.e.

µ(λ0) = Z 0

−σ

[1 +|λ0|(−s)]e^λ0sdV(ζ)(s) + Z 0

−τ

(−s)e^λ0sdV(η)(s).

Property (1.4) implies

0< µ(λ0)<1. (3.1)

We have

|γ(λ0)| ≤

Z 0

−σ

[1−λ0(−s)]e^λ0sdζ(s) +

Z 0

−τ

(−s)e^λ0sdη(s)

≤ Z 0

−σ

|1−λ0(−s)|e^λ0sdV(ζ)(s) + Z 0

−τ

(−s)e^λ0sdV(η)(s)

≤ Z 0

−σ

[1 +|λ₀|(−s)]e^λ0sdV(ζ)(s) + Z 0

−τ

(−s)e^λ0sdV(η)(s),

that is|γ(λ₀)| ≤µ(λ₀). So, in view of (3.1), it holds|γ(λ₀)|<1. This, in particular, implies that 1 +γ(λ0)>0.

Consider now an arbitrary initial function φ in C([−r,0],R) and let x be the solution of (1.1)-(1.2). Define

y(t) =e^−λ0tx(t) fort≥ −r.

Then, using the fact thatλ0is a (real) root of the characteristic equation (1.3), we obtain for everyt≥0

h x(t) +

Z 0

−σ

x(t+s)dζ(s)i0

− Z 0

−τ

x(t+s)dη(s)

=e^λ0tnh y(t) +

Z 0

−σ

e^λ0sy(t+s)dζ(s)i⁰ +λ₀h

y(t) + Z 0

−σ

e^λ0sy(t+s)dζ(s)i

− Z 0

−τ

e^λ0sy(t+s)dη(s)o

=e^λ0tnh y(t) +

Z 0

−σ

e^λ0sy(t+s)dζ(s)i0

+h

−λ₀ Z 0

−σ

e^λ0sdζ(s) + Z 0

−τ

e^λ0sdη(s)i y(t)

+λ0

Z 0

−σ

e^λ0sy(t+s)dζ(s)− Z 0

−τ

e^λ0sy(t+s)dη(s)o

=e^λ0tnh y(t) +

Z 0

−σ

e^λ0sy(t+s)dζ(s)i0

−λ0

Z 0

−σ

e^λ0s

y(t)−y(t+s) dζ(s)

+ Z 0

−τ

e^λ0s

y(t)−y(t+s) dη(s)o

.

Thus, sincexsatisfies (1.1) for all t≥0, it follows thaty satisfies h

y(t) + Z 0

−σ

e^λ0sy(t+s)dζ(s)i0

=λ0

Z 0

−σ

e^λ0s

y(t)−y(t+s) dζ(s)−

Z 0

−τ

e^λ0s

y(t)−y(t+s)

dη(s), t≥0.

(3.2)

On the other hand, the Initial Condition (1.2) becomes

y(t) =e^−λ0tφ(t) fort∈[−r,0]. (3.3) Furthermore, we can see that (3.2) is equivalently written as

y(t) + Z 0

−σ

e^λ0sy(t+s)dζ(s)

=λ0

Z 0

−σ

e^λ0shZ t t+s

y(u)dui dζ(s)−

Z 0

−τ

e^λ0shZ t t+s

y(u)dui

dη(s) +K fort≥0 for some real constant K. But, by taking into account (3.3) and the definition of L(λ0;φ), we have

K=y(0) + Z 0

−σ

e^λ0sy(s)dζ(s)−λ0

Z 0

−σ

e^λ0shZ 0 s

y(u)dui dζ(s)

+ Z 0

−τ

e^λ0shZ 0 s

y(u)dui dη(s)

=φ(0) + Z 0

−σ

φ(s)dζ(s)−λ0

Z 0

−σ

e^λ0shZ 0 s

e^−λ0uφ(u)dui dζ(s)

+ Z 0

−τ

e^λ0shZ 0 s

e^−λ0uφ(u)dui dη(s)

=φ(0) + Z 0

−σ

φ(s)−λ0e^λ0s Z 0

s

e^−λ0uφ(u)du

dζ(s) +

Z 0

−τ

e^λ0shZ 0 s

e^−λ0uφ(u)dui dη(s)

≡L(λ₀;φ).

So, (3.2) is equivalent to y(t) +

Z 0

−σ

e^λ0sy(t+s)dζ(s)

=λ₀ Z 0

−σ

e^λ0shZ t t+s

y(u)dui dζ(s)−

Z 0

−τ

e^λ0shZ t t+s

y(u)dui

dη(s) +L(λ₀;φ) (3.4)

fort≥0.

Next, we set

z(t) =y(t)− L(λ0;φ)

1 +γ(λ0) fort≥ −r.

Then, using the definition ofγ(λ0), it is easy to check that (3.4) takes the following equivalent form

z(t) + Z 0

−σ

e^λ0sz(t+s)dζ(s)

=λ0

Z 0

−σ

e^λ0shZ t t+s

z(u)dui dζ(s)−

Z 0

−τ

e^λ0shZ t t+s

z(u)dui

dη(s) fort≥0.

(3.5) Moreover, (3.3) is written as

z(t) =e^−λ0tφ(t)− L(λ₀;φ)

1 +γ(λ0) fort∈[−r,0]. (3.6) By the definitions ofy andz, what we have to prove is that

t→∞limz(t) = 0. (3.7)

In the rest of the proof we will establish (3.7). Put M(λ₀;φ) = max

t∈[−r,0]

e^−λ0tφ(t)− L(λ₀;φ) 1 +γ(λ0)

. Then, in view of (3.6), we have

|z(t)| ≤M(λ₀;φ) for −r≤t≤0. (3.8) We will show thatM(λ₀;φ) is a bound ofz on the whole interval [−r,∞), namely that

|z(t)| ≤M(λ0;φ) for allt≥ −r. (3.9) To this end, let us consider an arbitrary number >0. We claim that

|z(t)|< M(λ0;φ) + for everyt≥ −r. (3.10) Otherwise, because of (3.8), there exists a pointt0>0 such that

|z(t)|< M(λ0;φ) + for −r≤t < t0, and |z(t0)|=M(λ0;φ) +. Then, by taking into account the definition ofµ(λ0) and using (3.1), from (3.5) we obtain

M(λ₀;φ) +=|z(t₀)|

= −

Z 0

−σ

e^λ0sz(t0+s)dζ(s) +λ0

Z 0

−σ

e^λ0shZ t₀ t0+s

z(u)dui dζ(s)

− Z 0

−τ

e^λ0shZ t0

t₀+s

z(u)dui dη(s)

≤

Z 0

−σ

e^λ0sz(t0+s)dζ(s) +|λ0|

Z 0

−σ

e^λ0shZ t0

t0+s

z(u)dui dζ(s)

+

Z 0

−τ

e^λ0shZ t₀ t₀+s

z(u)dui dη(s)

≤ Z 0

−σ

e^λ0s|z(t0+s)|dV(ζ)(s) +|λ0| Z 0

−σ

e^λ0s

Z t0

t0+s

z(u)du

dV(ζ)(s) +

Z 0

−τ

e^λ0s

Z t₀

t₀+s

z(u)du

dV(η)(s)

≤ Z 0

−σ

e^λ0s|z(t0+s)|dV(ζ)(s) +|λ0| Z 0

−σ

e^λ0shZ t₀ t0+s

|z(u)|dui

dV(ζ)(s) +

Z 0

−τ

e^λ0shZ t₀ t₀+s

|z(u)|dui

dV(η)(s)

≤hZ 0

−σ

e^λ0sdV(ζ)(s) +|λ0| Z 0

−σ

e^λ0s(−s)dV(ζ)(s) +

Z 0

−τ

e^λ0s(−s)dV(η)(s)i

M(λ₀;φ) +

=nZ 0

−σ

[1 +|λ0|(−s)]e^λ0sdV(ζ)(s) + Z 0

−τ

(−s)e^λ0sdV(η)(s)o

[M(λ₀;φ) +]

≡µ(λ₀) [M(λ₀;φ) +]< M(λ₀;φ) +.

This is a contradiction and so our claim is true, i.e. (3.10) holds. We have thus proved that (3.10) is fulfilled for all numbers >0. Hence, (3.9) is satisfied. Now, by virtue of (3.9), from (3.5) we derive fort≥0,

|z(t)| ≤

Z 0

−σ

e^λ0sz(t+s)dζ(s) +|λ0|

Z 0

−σ

e^λ0shZ t t+s

z(u)dui dζ(s)

+

Z 0

−τ

e^λ0shZ t t+s

z(u)dui dη(s)

≤ Z 0

−σ

e^λ0s|z(t+s)|dV(ζ)(s) +|λ₀| Z 0

−σ

e^λ0s

Z t

t+s

z(u)du

dV(ζ)(s) +

Z 0

−τ

e^λ0s

Z t

t+s

z(u)du

dV(η)(s)

≤ Z 0

−σ

e^λ0s|z(t+s)|dV(ζ)(s) +|λ0| Z 0

−σ

e^λ0shZ t t+s

|z(u)|dui

dV(ζ)(s) +

Z 0

−τ

e^λ0shZ t t+s

|z(u)|dui

dV(η)(s)

≤hZ 0

−σ

e^λ0sdV(ζ)(s) +|λ0| Z 0

−σ

e^λ0s(−s)dV(ζ)(s) +

Z 0

−τ

e^λ0s(−s)dV(η)(s)i

M(λ₀;φ)

=nZ 0

−σ

[1 +|λ0|(−s)]e^λ0sdV(ζ)(s) + Z 0

−τ

e^λ0s(−s)dV(η)(s)o

M(λ0;φ).

Consequently, by the definition ofµ(λ0), we have

|z(t)| ≤µ(λ0)M(λ0;φ) for everyt≥0. (3.11) Using (3.5) and taking into account the definition of µ(λ0) as well as (3.9) and (3.11), one can show, by an easy induction, thatzsatisfies

|z(t)| ≤[µ(λ0)]^νM(λ0;φ) for allt≥νr−r (ν = 0,1,2, . . .). (3.12) Because of (3.1), we have lim_ν→∞[µ(λ₀)]^ν = 0. Thus, from (3.12) it follows that lim_t→∞z(t) = 0, i.e. (3.7) holds. The proof of Theorem 2.1 is complete.

4. Proof or Theorem 2.4

We first notice that, as in the proof of Theorem 2.1, we have 0 < µ(λ0) <1,

|γ(λ0)| ≤µ(λ0) and 1+γ(λ0)>0. It follows immediately thatN(λ0)>1. Consider an arbitrary functionφinC([−r,0],R) and letxbe the solution of (1.1)-(1.2). Let y andzbe defined as in the proof of Theorem 2.1, i.e.

y(t) =e^−λ0tx(t) fort≥ −r, and z(t) =y(t)− L(λ0;φ)

1 +γ(λ₀) fort≥ −r, whereL(λ₀;φ) is defined as in Theorem 2.1. Moreover, let M(λ₀;φ) be defined as in the proof of Theorem 2.1, i.e.

M(λ₀;φ) = max

t∈[−r,0]

e^−λ0tφ(t)− L(λ₀;φ) 1 +γ(λ0)

.

Then, as in the proof of Theorem 2.1, we can show thatz satisfies (3.11), namely

|z(t)| ≤µ(λ₀)M(λ₀;φ) for everyt≥0.

By the definition ofz, from the last inequality it follows that

|y(t)| ≤ |L(λ0;φ)|

1 +γ(λ₀)+µ(λ0)M(λ0;φ) fort≥0. (4.1) On the other hand, from the definition ofM(λ₀;φ) we get

M(λ₀;φ)≤ kφkmax{1, e^λ0r}+|L(λ₀;φ)|

1 +γ(λ0). So, (4.1) gives

|y(t)| ≤ 1 +µ(λ₀)

1 +γ(λ0)|L(λ0;φ)|+kφkµ(λ0) max{1, e^λ0r}, t≥0. (4.2) Furthermore, by the definition ofL(λ0;φ), we obtain

|L(λ0;φ)| ≤ |φ(0)|+

Z 0

−σ

h

φ(s)−λ0e^λ0s Z 0

s

e^−λ0uφ(u)dui dζ(s)

+

Z 0

−τ

e^λ0shZ 0 s

e^−λ0uφ(u)dui dη(s)

=|φ(0)|+

Z 0

−σ

he^−λ0sφ(s)−λ₀ Z 0

s

e^−λ0uφ(u)dui

e^λ0sdζ(s)

+

Z 0

−τ

hZ 0

s

e^−λ0uφ(u)dui

e^λ0sdη(s)

≤ |φ(0)|+ Z 0

−σ

e^−λ0sφ(s)−λ0

Z 0

s

e^−λ0uφ(u)du

e^λ0sdV(ζ)(s) +

Z 0

−τ

Z 0

s

e^−λ0uφ(u)du

e^λ0sdV(η)(s)

≤ |φ(0)|+ Z 0

−σ

h

e^−λ0s|φ(s)|+|λ0| Z 0

s

e^−λ0u|φ(u)|dui

e^λ0sdV(ζ)(s) +

Z 0

−τ

hZ 0

s

e^−λ0u|φ(u)|dui

e^λ0sdV(η)(s). Consequently

|L(λ0;φ)| ≤ kφkh 1 +

Z 0

−σ

e^−λ0s+|λ0| Z 0

s

e^−λ0udu

e^λ0sdV(ζ)(s) +

Z 0

−τ

Z 0

s

e^−λ0udu

e^λ0sdV(η)(s)i .

(4.3)

We have previously used the elementary inequality e^−λ0t≤max{1, e^λ0r} for each t∈[−r,0]. Therefore,

e^−λ0s≤max{1, e^λ0r} fors∈[−σ,0], Z 0

s

e^−λ0udu≤(−s) max{1, e^λ0r} fors∈[−σ,0], Z 0

s

e^−λ0udu≤(−s) max{1, e^λ0r} fors∈[−τ,0].

Thus, (4.3) leads to

|L(λ0;φ)| ≤ kφkn

1 +Z 0

−σ

[1 +|λ0|(−s)]e^λ0sdV(ζ)(s) +

Z 0

−τ

(−s)e^λ0sdV(η)(s)

max{1, e^λ0r}o , which, in view of the definition ofµ(λ₀), can be written as

|L(λ0;φ)| ≤ kφk

1 +µ(λ₀) max{1, e^λ0r} . Hence, fort≥0, (4.2) gives

|y(t)| ≤n1 +µ(λ0) 1 +γ(λ0)

1 +µ(λ0) max{1, e^λ0r}

+µ(λ0) max{1, e^λ0r}o kφk

=n1 +µ(λ0) 1 +γ(λ₀)+h

1 + 1 +µ(λ0) 1 +γ(λ₀) i

µ(λ0) max{1, e^λ0r}o kφk and so, because of the definition ofN(λ0), we have

|y(t)| ≤N(λ₀)kφk for everyt≥0.

Finally, in view of the definition ofy, we obtain

|x(t)| ≤N(λ0)kφke^λ0t for allt≥0. (4.4) This completes the proof of the first part of the theorem. It remains to show the stability criterion contained in the theorem.

Let us suppose thatλ0≤0. Letφ∈C([−r,0],R) be an arbitrary initial function and letxbe the solution of (1.1)-(1.2). Then (4.4) holds and hence

|x(t)| ≤N(λ0)kφk for every t≥0.

Since N(λ₀) > 1, it follows that |x(t)| ≤ N(λ₀)kφk for all t ≥ −r. Using this inequality, we can immediately verify that the trivial solution of (1.1) is stable (at 0). Moreover, ifλ0<0, then (4.4) guarantees that

t→∞lim x(t) = 0.

Thus, forλ0<0 the trivial solution of (1.1) is asymptotically stable (at 0). Because of the autonomous character of (1.1), the trivial solution of (1.1) is uniformly stable ifλ0= 0 and it is uniformly asymptotically stable ifλ0<0.

Finally, we assume that λ0 > 0 and we will show that the trivial solution of (1.1) is unstable. Suppose, for the sake of contradiction, that the trivial solution of (1.1) is stable (at 0). Then we can choose a number δ >0 such that, for each φ∈C([−r,0],R) withkφk< δ, the solution xof (1.1)-(1.2) satisfies

|x(t)|<1 for allt≥ −r. (4.5) Set

φ0(t) =e^λ0t fort∈[−r,0].

We see thatφ₀∈C([−r,0],R) and, as in Section 2, we can verify that

L(λ₀;φ₀) = 1 +γ(λ₀)>0, (4.6) whereγ(λ0) and, for anyφ∈C([−r,0],R),L(λ0;φ) are defined as in Theorem 2.1.

Next, we consider a numberδ0 with 0< δ0< δ and we put φ= δ0

kφ₀kφ0.

Clearly,φbelongs toC([−r,0],R) andkφk=δ0< δ. Hence, for this initial function, the solutionxof (1.1)-(1.2) satisfies (4.5). On the other hand, by applying Theorem 2.1 and taking into account (4.6) as well as the linearity of the operator L(λ0;·), we obtain

t→∞lim

e^−λ0tx(t)

= L(λ0;φ)

1 +γ(λ₀) = (δ0/kφ0k)L(λ0;φ0) 1 +γ(λ₀) = δ0

kφ₀k >0.

But, sinceλ0>0, from (4.5) it follows that

t→∞lim

e^−λ0tx(t)

= 0.

We have thus arrived at a contradiction. The proof of Theorem 2.4 is now complete.

5. Application of the main results to the special case of non-neutral equations

In this section, we will concentrate on the (non-neutral) delay differential equa- tion (1.5) and we shall apply our main results to this equation. For the delay differential equation (1.5), the following results hold.

Theorem 5.1. Let λ0 be a real root of the characteristic equation (1.7)with the property (1.8). Then, for anyφ∈C([−τ,0],R), the solutionxof (1.5)-(1.6)satis- fies

t→∞lim

e^−λ0tx(t)

= `(λ0;φ) 1 +R0

−τ(−s)e^λ0sdη(s), where

`(λ0;φ) =φ(0) + Z 0

−τ

e^λ0shZ 0 s

e^−λ0uφ(u)dui dη(s).

Note that Property (1.8) guarantees that 1 +R0

−τ(−s)e^λ0sdη(s)>0.

Corollary 5.2. Assume that

η(−τ) =η(0) and Z 0

−τ

(−s)dV(η)(s)<1. (5.1) Then, for anyφ∈C([−τ,0],R), the solution xof (1.5)-(1.6)satisfies

t→∞lim x(t) = φ(0) +R0

−τ

R0

s φ(u)du dη(s) 1 +R0

−τ(−s)dη(s) . Note that the second assumption of (5.1) ensures that 1 +R0

−τ(−s)dη(s)>0.

Corollary 5.3. Let λ0 be a real root of the characteristic equation (1.7) with the property (1.8). Then, for anyφ∈C([−τ,0],R), the solutionxof (1.5)-(1.6)will be nonoscillatory, except possibly if φ satisfies `(λ0;φ) = 0, where `(λ0;φ)is defined as in Theorem 5.1.

As a complement to Corollary 5.3, we have: Let λ0 be a real root of the charac- teristic equation (1.7)with the property (1.8). Moreover, for anyφ∈C([−τ,0],R), let `(λ0;φ) be defined as in Theorem 5.1. Then the set of all functions φ ∈ C([−τ,0],R) which satisfy `(λ0;φ) = 0 is a nowhere dense subset of the Banach spaceC([−τ,0],R) (with the sup-norm).

Theorem 5.4. Let λ0 be a real root of the characteristic equation (1.7)with the property (1.8). Then, for anyφ∈C([−τ,0],R), the solutionxof (1.5)-(1.6)satis- fies

|x(t)| ≤n(λ₀)kφke^λ0t for allt≥0, where

n(λ0) = 1 +R0

−τ(−s)e^λ0sdV(η)(s) 1 +R0

−τ(−s)e^λ0sdη(s) +h

1 +1 +R0

−τ(−s)e^λ0sdV(η)(s) 1 +R0

−τ(−s)e^λ0sdη(s)

ihZ 0

−τ

(−s)e^λ0sdV(η)(s)i

max{1, e^λ0τ} with the constantn(λ₀)being greater than 1. Moreover, the trivial solution of (1.5) is uniformly stable ifλ₀= 0, uniformly asymptotically stable ifλ₀<0, and unstable if λ₀>0.

We observe that, concerning the uniform stability, the corresponding result in Theorem 5.4 can be equivalently stated as: The trivial solution of (1.5)is uniformly stable if Condition (5.1)holds.

6. Sufficient conditions for the characteristic equation to have a real root with the property required

In this section, we give some conditions, under which the characteristic equation (1.3) (and, in particular, the characteristic equation (1.7)) has a real rootλ₀ with the property (1.4) (and, in particular, with the property (1.8)).

Lemma 6.1. Assume that Z 0

−σ

e^−s/rdζ(s) +r Z 0

−τ

e^−s/rdη(s)>−1, (6.1)

− Z 0

−σ

e^s/rdζ(s) +r Z 0

−τ

e^s/rdη(s)<1, (6.2) Z 0

−σ

[1 + (−s)/r]e^−s/rdV(ζ)(s) + Z 0

−τ

(−s)e^−s/rdV(η)(s)≤1. (6.3) Then, in the interval (−1/r,1/r), the characteristic equation (1.3) has a unique rootλ0, and this root satisfies the property (1.4).

Proof. Define F(λ) =λh

1 + Z 0

−σ

e^λsdζ(s)i

− Z 0

−τ

e^λsdη(s) forλ∈[−1/r,1/r].

We have

F(−1/r) =−1 r h

1 + Z 0

−σ

e^−s/rdζ(s)i

− Z 0

−τ

e^−s/rdη(s)

=−1 r h

1 + Z 0

−σ

e^−s/rdζ(s) +r Z 0

−τ

e^−s/rdη(s)i

and so, by (6.1), we getF(−1/r)<0. Moreover, F(1/r) = 1

r h

1 + Z 0

−σ

e^s/rdζ(s)i

− Z 0

−τ

e^s/rdη(s)

=−1 r

h−1− Z 0

−σ

e^s/rdζ(s) +r Z 0

−τ

e^s/rdη(s)i

and hence from (6.2) it follows that F(1/r) > 0. Furthermore, by taking into account (6.3), forλ∈(−1/r,1/r), we obtain

F⁰(λ) = 1 + Z 0

−σ

[1−λ(−s)]e^λsdζ(s) + Z 0

−τ

(−s)e^λsdη(s)

≥1−

Z 0

−σ

[1−λ(−s)]e^λsdζ(s)

−

Z 0

−τ

(−s)e^λsdη(s)

≥1− Z 0

−σ

|1−λ(−s)|e^λsdV(ζ)(s)− Z 0

−τ

(−s)e^λsdV(η)(s)

≥1− Z 0

−σ

[1 +|λ|(−s)]e^λsdV(ζ)(s)− Z 0

−τ

(−s)e^λsdV(η)(s)

>1− Z 0

−σ

[1 + (−s)/r]e^−s/rdV(ζ)(s)− Z 0

−τ

(−s)e^−s/rdV(η)(s)

≥0.

Therefore, F is strictly increasing on the interval (−1/r,1/r). So, in the interval (−1/r,1/r), the equation F(λ) = 0 (which coincides with (1.3)) has a unique root λ0. This root satisfies (1.4). Indeed, by using again (6.3), we have

Z 0

−σ

[1 +|λ0|(−s)]e^λ0sdV(ζ)(s) + Z 0

−τ

(−s)e^λ0sdV(η)(s)

<

Z 0

−σ

[1 + (−s)/r]e^−s/rdV(ζ)(s) + Z 0

−τ

(−s)e^−s/rdV(η)(s)≤1.

This completes the proof.

Now, we will confine our attention to the special case of the (non-neutral) delay differential equation (1.5), for which the characteristic equation is (1.7). In this case, Conditions (6.1), (6.2), (6.3) take the form

τ Z 0

−τ

e^−s/τdη(s)>−1, (6.4)

τ Z 0

−τ

e^s/τdη(s)<1, (6.5)

Z 0

−τ

(−s)e^−s/τdV(η)(s)≤1. (6.6)

Lemma 6.1 can be applied to the case of the characteristic equation (1.7) with the assumptions (6.4)–(6.6) instead of (6.1)–(6.3). However, we have the following result which is slightly better.

Lemma 6.2. Let (6.4)and (6.6)be satisfied. Then, in the interval(−1/τ,∞), the characteristic equation (1.7)has a unique root λ0; this root has the property (1.8) and, provided that (6.5)holds, the rootλ0 is less than 1/τ.

Proof. Set

F0(λ) =λ− Z 0

−τ

e^λsdη(s) forλ≥ −1/τ.

From (6.4), it follows immediately that F0(−1/τ)<0. Next, for everyλ≥ −1/τ, we obtain

F0(λ)≥λ−

Z 0

−τ

e^λsdη(s) ≥λ−

Z 0

−τ

e^λsdV(η)(s)≥λ− Z 0

−τ

e^−s/τdV(η)(s) and consequentlyF0(∞) =∞. Moreover, forλ >−1/τ, we have

F₀⁰(λ) = 1 + Z 0

−τ

(−s)e^λsdη(s)≥1−

Z 0

−τ

(−s)e^λsdη(s)

≥1− Z 0

−τ

(−s)e^λsdV(η)(s)>1− Z 0

−τ

(−s)e^−s/τdV(η)(s)

and so, by (6.6), it follows that F0 is strictly increasing on (−1/τ,∞). Hence, in the interval (−1/τ,∞), there exists a unique rootλ₀of the equationF₀(λ) = 0 (or, equivalently, of (1.7)). By using again (6.6), we get

Z 0

−τ

(−s)e^λ0sdV(η)(s)<

Z 0

−τ

(−s)e^−s/τdV(η)(s)≤1.

Consequently the root λ0 satisfies (1.8). Now assume that (6.5) is also satisfied.

This assumption implies that F0(1/τ) > 0. Thus, we can immediately conclude that the rootλ0 is always less than 1/τ. The proof is now complete.

We remark that Conditions (6.4)–(6.6) are satisfied if the following stronger condition holds:

τ Z 0

−τ

e^−s/τdV(η)(s)<1. (6.7)

In fact, we have τ

Z 0

−τ

e^−s/τdη(s)≥ −τ

Z 0

−τ

e^−s/τdη(s) ≥ −τ

Z 0

−τ

e^−s/τdV(η)(s),

τ Z 0

−τ

e^s/τdη(s)≤τ

Z 0

−τ

e^s/τdη(s) ≤τ

Z 0

−τ

e^s/τdV(η)(s)

≤τ Z 0

−τ

dV(η)(s)≤τ Z 0

−τ

e^−s/τdV(η)(s) and

Z 0

−τ

(−s)e^−s/τdV(η)(s)≤τ Z 0

−τ

e^−s/τdV(η)(s) and so our assertion is true. Furthermore, since

τ Z 0

−τ

e^−s/τdV(η)(s)≤τ e Z 0

−τ

dV(η)(s) =τ eV(η)(0), we conclude thatCondition (6.7)holds if

τ eV(η)(0)<1. (6.8)

Note thatV(η)(0) is the total variation ofη on the interval [−τ,0]. Condition (6.7) and, in particular, Condition (6.8) were used by Driver [3].

Note that it is an interesting question to find other conditions onσ andτ and on the integratorsζandη, which are sufficient for the characteristic equation (1.3) to have a real rootλ₀ with the property (1.4). This problem remains interesting still in the special case of the characteristic equation (1.7).

Before closing this section and the paper, we will use Lemma 6.1 (and, in par- ticular, Lemma 6.2) to find some explicit conditions in terms ofσ,τandζ,η (and, in particular, in terms of τ and η), under which the trivial solution of (1.1) (and, in particular, of (1.5)) is uniformly asymptotically stable or unstable. Note that analogous conditions for the uniform stability of the trivial solution of (1.1) (and, in particular, of (1.5)) have already been given in previous sections.

Let us assume that (6.1)–(6.3) hold. Then Lemma 6.1 guarantees that, in the interval (−1/r,1/r), the characteristic equation (1.3) has a unique root λ0; this root satisfies the property (1.4). LetF be defined as in the proof of Lemma 6.1.

For this function, as in the proof of Lemma 6.1, we have F(−1/r)<0 and F(1/r)>0.

Clearly, λ0 is negative if F(0) >0, and λ0 is positive if F(0)< 0. On the other hand,

F(0) =− Z 0

−τ

dη(s) =−[η(0)−η(−τ)].

So,λ0<0 if η(0)< η(−τ), andλ0>0 if η(0)> η(−τ). Hence, from the stability criterion contained in Theorem 2.4 we can obtain the following result.

Corollary 6.3. Assume that (6.1)–(6.3) are satisfied. Then the trivial solution of (1.1) is uniformly asymptotically stable if η(0) < η(−τ) and it is unstable if η(0)> η(−τ).

By an analogous way, we can use Lemma 6.2 and the stability criterion contained in Theorem 5.4 to derive the following result.

Corollary 6.4. Assume that (6.4)and(6.6)are satisfied. Then the trivial solution of (1.5) is uniformly asymptotically stable if η(0) < η(−τ) and it is unstable if η(0)> η(−τ).

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Department of Mathematics, University of Ioannina, P.O. Box 1186, 451 10 Ioannina, Greece

E-mail address, Christos G. Philos: [email protected] E-mail address, Ioannis K. Purnaras: [email protected]